Peter is a lively lad. All systems are at ‘go’. He will usually eat when he’s hungry, like any healthy boy.

There was no trouble about lunch last Saturday. It was roast chicken with bread sauce—one of his favourites. He tackled it with relish, and we were smoothly through to pudding at a brisk rate of knots.

But today things are different. He’s hungry all right, and it’s roast chicken again. But he won’t make a start. Why not? Now that he’s six he knows what he likes, and Mum has forgotten the bread sauce. Would you believe it? But it’s too late to make some now, so we shall all have to be satisfied without it.

The article by Margaret E. Baron on the historical development of logic diagrams (The Mathematical Gazette, LIII, May 1969) aroused my interest in the infeasibility of drawing a diagram extendable to n classes. The difficulty of following Venn’s suggestion of a comb-like structure is shown in the complexity of drawing such a diagram for five classes.

Numbers of the form n(n + 1)/2 are called triangular numbers. This is because and the left-hand side of this equation can be represented by a triangle as in Fig. (a). Numbers of the form n are called square numbers, or simply squares, for obvious similar reasons. We can also have rectangular numbers, but there is little point in defining them as numbers of the form m . n, since then they will include all the numbers. Instead we define a rectangular number as one which is of the form n(n + 1) —that is, the rectangle concerned must have one side exceeding the other by exactly one unit.

The subject of this address is Technical Education. I wish to examine its essential nature and also its relation to a liberal education. Such an enquiry may help us to realise the conditions for the successful working of a national system of technical training. It is also a very burning question among mathematical teachers; for mathematics is included in most technical courses.

Suppose you are trying to tile an infinite grid of unit squares with 2 x 1 rectangles so that each rectangle covers two squares. You place two of the rectangles without thinking. Can the tiling now be accomplished? As the answer is “yes” I call the rectangle an “idiot-proof tile”. In general a polyomino will be called idiot proof if the infinite grid can be tiled irrespective of the positions of two of the tiles (the idiotic tiles), except that the idiotic tiles shall not be overlapping and shall not completely surround an area that is less than that of the tile (not even an idiot would do that!). The first interesting idiot-proof tile (IPT) is the L-tromino.

Bayesian statistics is included in few elementary statistics courses, and many mathematicians have heard of it, perhaps through collateral readings from popular literature or [1], selected as an Editor's Choice in the New York Times Book Review. ‘Bayesian statistics’ provides for a way to incorporate prior beliefs, experience, or information into the analysis of data. Bayesian thinking is natural, and that is an advantage. For example, on a summer morning, if we see dark rain clouds up in the sky, we leave home for work with an umbrella because prior experience tells us that doing so is beneficial. In general, the idea is simple; schematically, it looks like this:

(prior belief) + (data: new information) ⇒ (posterior belief).

Thus, we begin with a prior belief that we allow to be modified or informed by new data to produce a posterior belief, which then becomes our new prior, and this process is never-ending. We are always willing to update our beliefs according to new information.

The evaluation of moments of inertia of simple solids is frequently regarded merely as a convenient source of problems in the integral calculus. From the point of view of the student of dynamics it is, however, perhaps even more instructive to make these calculations without any actual integration, for in this way he may obtain a deeper insight into the physical significance of the underlying concept. Whether this be so or not it seemed to the writer an amusing task (performed during a moment of inertia caused by a bacteriological infection) to see how many of the well known formulae could be derived by such methods.

This perennial problem has led to many articles in the Gazette (listed by Sullivan in the June 1987 edition) many of them geometric or combinatorial in nature. If we look at some older algebra texts, published before the 1960s say, we see that the method of differences is one of the standard methods for dealing with the problem. With the current interest in discrete mathematics this method merits re-examination. A feature of the method is its effectiveness in finding the solution to the problem for all powers and it is not limited to the sums of squares and cubes. In this article we hope to show that it can be treated more transparently using the calculus of finite differences.